The Shape of Shapes: An Ontological Exploration
Robert J. Rovetto
University at Buffalo, The State University of New York
Department of Philosophy
[email protected]
[email protected]
Abstract: What are shapes? The ancient Greek geometer is perhaps the most familiar with
these entities, but shape is general enough to transcend domains of inquiry. In this
communication I present a broad examination of shape and related notions, such as form,
boundary and surface. I aim to explore this question and describe the category of shape by
identifying some of its most general features. I contrast geometric (or perfect) shapes with
physical or organic shapes (the shape of objects), discuss granularity, artifactuality, the
cognitive aspect of shape and introduce a preliminary ontological hierarchy of shape.
Keywords: ontology, shape, form, boundary, physical discontinuity, mathematical entity
1.0 Introduction
What are shapes? The ancient Greek geometer is perhaps the most acquainted with
the notion of shape, having made geometry into an abstract discipline, while keeping it ―firmly
rooted in the real world where it could give him true and reliable information about actual
spatial relationships.‖[1]. Yet all of us have an implicit idea of shape in our mind‘s eye,
whether from education in geometry or from our inherent ability to recognize recurrent and
similar forms and figures in the world. In this communication I aim to explore this question
and describe our notion of shape. I identify some general features of shapes, and determine
their ontological category. The notions of form, boundary, surface and granularity are
introduced in relation to shape.
An important distinction to keep in mind is that between ideal, perfect and abstract
geometric shapes on the one hand, and imperfectI, physical or organic mind-external shapes
on the other. Call the former ‗geometric shapes‘ and the latter ‗physical shapes‘ or ‗organic
shapes‘. This distinction can be understood as being parallel to types (classes, universals,
general entities) and instances (individuals or particulars in the world). Geometric shapes
typically have precise mathematical formalizations. Their exact physical manifestations are
not, so far as I am aware, observed in mind-external reality, only approximated by entities
exhibiting a similar shape. In this sense geometric shapes are idealizations or abstractions.
This makes geometric shapes similar to types or universals. Their instances are inexact
replicas of the shape type in question, but have similar attributes or properties in common,
properties characterizing the type.
By contrast, organic or physical shapes are irregular or uneven shapes of mindexternal objects or things in the world. A planet is not perfectly spherical, and the branches of
I
In using the words ‗perfect‘ or ‗imperfect‘ I do not use them in any normative sense.
a tree are not perfectly cylindrical, for example. ‗Perfectly‘ is used here in the sense of
coinciding with or physically manifesting the exact mathematical definitions, or precise
symmetrical relations, of geometric shapes. Objects and physical phenomena in the world,
rarely if ever, manifest or exhibit any concretization of geometric shapes, but this is not to say
that it is not possible or that it does not obtain at times. Objects are not precisely symmetrical
about a given axis, cube-shaped things do not have faces of exactly the same area, for
example, and there is no concretization of a perfect sphere. Note that the terms ‗object‘ and
‗material object‘ are used in the sense of ‗object‘ in [2]. The term ‗entity‘ is used in its usual
way, as a catch-all for anything that exists, from objects to processes to thoughts.
I begin with a discussion of the dimensionality of shapes in section 2.0, followed by
an ontological categorization of shape in 3.0. Descriptions and definitions of ‗form‘ and
‗shape‘ are introduced in 4.0, boundary and surface are introduced in 5.0, granularity in 6.0,
and a simple ontology in 7.0. Section 8.0 ends with my closing remarks and reflections.
Throughout I enumerate some plausible features and descriptions of shape.
2.0 Dimensionality of Shape
It seems clear that shapes are not to be found in the realm of the zero-dimensional.
There are no zero-dimensional shapes, for that which is zero-dimensional, although an entity
in the arsenal of the mathematician, is but an extensionless point. From an abstract
perspective, points can be understood as parts of (higher-dimensional) shapes. For example,
for every line segment there are two points bounding its extent. Although points in space
appear natural, they are abstractions which cannot be materialized [3].
Extending to the one-dimensional (1D for short), we can say that shapes come into
being. Examples are lines, including straight lines and line segments, rays, sinusoids,
parabolas, and so forth. If we understand lines and other geometric entities as being introduced
by ancient mathematicians to represent more-or-less straight objects in the world and, thus, as
idealizations or abstractions of mind-external objects,II then we are better able to view these
1D entities as geometric shapes.
The notion of shape connotes extension from both perspectives of physical space (as
in mind-external), and mental, conceptual or abstract space (as in mind-internal).III The latter
is to be understood as the geometer‘s abstract realm of perfect shapes that have, or could have,
precise formulations. To avoid confusions, we may call this realm ‗geometric space‘. It is here
that the geometer manipulates her objects of inquiry and it is to the Earth (and to spatiality,
among other things) that the geometer applies and projects her tools and domain-specific
objects (shapes). Perhaps shape is a spatial form, a subtype of the more general type form.
Note that in using the words ‗abstract‘ and ‗realm‘ I do not suggest the literal existence of an
abstract world divorced from our spatiotemporal universe. I make no commitment to the
mind-external reality of mathematical entities, but an inquiry into the ontology of shape will
require exploration into the philosophy of mathematics. This will involve inquiry into the
II
With regard to lines: http://en.wikipedia.org/wiki/Line_%28geometry%29. It is reasonable that
geometric entities were used, posited or introduced by the ancients as representations of objects in the
world. That geometry has been practically used (art, navigation and surveying, etc.) since antiquity, and
that it likely started as a practical enterprise, attests to that.
III
This is not to say that shapes literally occupy our minds (as if they were physical occupants), only that
abstract and perfect geometric shapes are not found outside of the mind.
ontological status of mathematical entities and the relation of mathematics and other formal
expressions to the world. At the very least and in some sense, mathematical entities exist in a
cognitive fashion (perhaps as representations), in minds as ideas. Although I am skeptical, I
leave open the question as to their mind-independent existence. Mathematics, in conjunction
with its applicable expression in physics, is the formal language we use to communicate order,
structure, pattern, and (nomological) principledness in the universe.
These geometric entities reflect the ideal and perfect shapes we are able to conceive,
mathematically formalize, identify in (some say discover) and project onto the imperfect
(although structured) spatio-temporal world of objects, artifacts, landscapes, astronomical
phenomena and geographic entities. Practical examples of projecting geometric shapes onto
the mind-external world include reference shapes in global positioning systems, coastal
navigation and surveying. It is very likely that the development of geometry began with
solving practical problems in mind.
We have, then, the following. (S1a) Shapes are necessarily extended. Alternatively,
(S1b) If an entity has or exhibits a shape then that entity is necessarily spatially extended, and
(S2) Shapes are at least one-dimensional (Shapes have at least one-dimensional extent), or
alternatively, If an entity has or exhibits a shape, then that entity is at least one-dimensional.IV
In (S1a) I make no mention of the kind of extension—spatial, temporal, or otherwise—
because we can understand shapes as spatially and materially extended in mind-external
reality, and as extended in a non-physical or immaterial sense in conceptual, cognitive, or
mind-internal reality. Representations of shape may fall under one of these types of extension.
Two-dimensionality (2D for short) presents us with some of the most familiar types
of shape: the circle, the triangle and its subtypes (equilateral, isosceles, right, scalene), the
square, the hexagon, and so forth. Here is where our basic studies in geometry find a home.
Here is where we comprehend points, sides (or edges), angles, congruence, and symmetry. As
we vary the length of one or more sides of symmetric or equilateral 2D shapes, we form a
hodgepodge of asymmetrical shapes. Likewise, as we vary the curvature of curved shapes, all
sorts of unique and irregular figures are born. It quickly becomes apparent that there are
infinitely many shapes in the geometer‘s toolbox: (S3) There are infinitely many geometric
shapes. This applies to two- three- and higher-dimensional shapes alike.
Under a given geometry of space—Euclidean or Non-Euclidean, for example—2D
shapes can be formally described. The geometric descriptions inform us of their properties,
such as having a certain number (quantity) of sidesV, and the number and measure of angles
formed by those sides. These descriptions may vary depending on the adopted geometry.
Using spherical geometry, the sum of the angles of a triangle is greater than that of a triangle
in Euclidean geometry.VI The specific length of the sides, however, is not described.
Increasing or decreasing the length of the sides of a square, for example, will not cause it to
cease to be a square so long as (a) the lengths of all the sides remain equal, and (b) the
dimensionality remains constant. In other words, if the identity of the shape as a square
persists, then any decrease in the length of all the four sides does not result in a dimensional
IV
If the reader is very uncomfortable with the idea of lines, sinusoids and other 1D entities being
classified as shapes, then two-dimensionality may be considered the minimum dimensionality in which
shapes take shape. Such a change has no significant effect on this communication.
V
Shapes can be classified by the number of sides they have. For the three-dimensional polyhedra, they
can be classified according to the number of faces.
VI
Transcending the specific geometry, the general notion of triangle (or triangular) remains.
collapse or dimensional descent into a (one-dimensional) point. It may be asserted that any
specificity in the measure of the sides of a shape indicates the presence of an individual shape,
a shape instance. This is not obvious, however, because abstract geometric shapes can be
constructed without their manifestation obtaining (as the particular shape of an object in the
world). Each side of a 2D shape can be considered one of its parts.
Having four connected sides of equal length is an essential property of the square, as
is having four angles of ninety degrees formed by the meeting of the sides. If a shape loses
one of its sides, then it also loses its identity as that type of shape. A square that loses one of
its four sides is no longer a square. If we consider sides to be parts of shapes, employing either
a broad sense of parthood for material and immaterial entities, or a specific (say, immaterial)
sense of parthood, then we exclude some ontological categorizations of shape. For example,
we rule out independent continuant [2] as the ontological category of shape because
independent continuants retain their identity while losing or gaining parts.VII It follows that
object is ruled out as well because it is a subtype of independent continuant. Although shapes
are a unity that is self-contained and extended, it is not at all clear that they are material
entities so object is not an appropriate ontological classification. In a similar vein, individuals
can gain and lose parts [4], but if we are presented with an individual (particular) shape and
remove one of its sides it no longer persists as the same individual. No longer is it an instance
of its type. To avoid excluding shapes as individuals—essentially excluding shape instances—
one can argue that shapes do not have parts. Shapes that lose parts in both geometric space and
in physical space seem to lose their identity, however. Finally, it makes little sense to say that
the square changes into a line, but it does make sense to speak of the shape of an object
changing. This fact is reflected in an account of shapes as properties of objects that undergo
change over time.
When we arrive at three-dimensionality (3D for short) we arrive at our world of
everyday objects. The objects in this dimensionality are expressed as having length, width,
and height. In the abstract three-dimensional realm of the geometer we form and describe
entities such as cubes, spheres, cylinders, and rectangular pyramids. These are all threedimensional geometric shapes. Here, the geometer can describe 3D shapes in terms of lowerdimensional shapes. A cube has six 2D faces for example. In using the term ‗side‘, we say: a
1D side is the shape itself, a 2D side is an edge, and a 3D side is a face (or simply side).
Comparing these ideal 3D entities with objects in the physical world, we find oranges that
have a roughly spherical shape, tree trunks approximating cylinders, and so forth.
In our 3D spatio-temporal universe, however, matters become more complicated
because we no longer are solely considering the perfect abstract shapes of the geometer. We
are now considering the shape of things. We are describing objects in the world as having a
shape, and we observe similarity-of-shape among objects. Further, physics, physical
geometry, and physical space-time all are related. Objects, organisms, and even aggregate-like
entities such as clouds have the shape they do in virtue of a number of factors, all of which
work together to present the shapes we observe, recurrently recognize, and (occasionally)
formalize. Some of these factors include environmental processes, biological processes, and
processes internal to the object. These factors, in turn, have a nomological foundation
affording recurrence, similarity of shape and attribute-agreement in general. In this way, I am
VII
We also rule out shapes as independent continuants because the latter do not inhere in any entity. As
we explore further we see that shapes are likely properties, and properties are said to inhere in other
entities.
inclined toward understanding many types or universalsVIII in terms of nomological factors
rather than vice versa.
If universals are our way of explaining similarity, recurrence, repetition, and
attribute-agreement in the world, then these latter aspects are, in turn, the result of the
principled structure of the world. I am resistant to the idea that all there is to an ontological
account of shape (and other categories) is that shape types exist in nature and that shape
instances inhere in objects. Rather, the physical shape of an object is due to the complex
network of naturally principled and physical influences involved in the existential becoming
(or creation, if you will) of the object exhibiting that shape.IX Since ontology (applied and
philosophical) traditionally makes use of types and individuals, universals and instances, the
task is to produce an account of shape in these terms. This is particularly the case if it is true
that ―the appropriate subdivision of continuous reality is that into specific types of
individuals‖[3]. It is, nevertheless, accurate to understand many types or universals in terms
of, or as describingX, (i) physical principles or laws of nature underlying (ii) the abovementioned lower-level elements (phenomena in the world). In other words, the explanation of
shape as an ontological category involves numerous elements and considerations.
3.0 Shapes and The Shaped: Shapes as Properties and Shapes as Objects
With respect to the mind-external world, notice that if shapes are properties (of
things), then we may have a situation in which properties have properties. At first glance this
seems true because we predicate shape of objects in the world; we say that objects have a
certain shape. We also describe types of shapes as having specific properties. If a shape is
defined as having a particular number of sides (as with polygons), a particular curvature (as
with curved shapes, such as the circle and the ellipse), specific relations between sides, or
otherwise, then it should be apparent that we are describing properties of properties of things.
We might be inclined to say that it is the shape that has a certain amount of angles and sides,
rather than the object bearing the shape in question, but this is not entirely accurate. Shapes,
conceived as objects in their own right (in geometric space), have sides, but in our
spatiotemporal world, objects have sides, and surfaces, as well. When we divorce the shape
from that which has the shape via abstraction, we use ‗side‘ for the former as much as we do
for the latter. The distinction between geometric and physical space, between ideas and ideal
or cognitive constructions and material mind-external particulars is significant.
In the abstract geometric space of the geometer, we conceive or speak of shapes as if
they were objects in their own right and identify (perhaps, prescribe) their properties. We say
‗as if‘ because the term ‗object‘ is intuitively applied to material entities in the world. Relative
to the geometer‘s domain it is the case that shapes have particular properties, but here ‗object‘
implicitly takes on a broader meaning such that there are geometric objects. The geometer‘s
toolbox contains infinitely many of these so called objects. A triangle, we may say, is a
polygon that has three angles formed by three connected edges at the endpoint of each edge,
the sum of the angles being 180 (relative to Euclidean geometry, of course). A square is a
polygon that has four right angles formed by four connected sides of equal length.
VIII
If only in the philosophical sense.
An account of geometric shapes may not be so easily stated, however.
X
In a unified if compartmentalized manner
IX
The fact that descriptions of geometric shapes do not specify the length of the sides
demonstrates the generality and abstract nature of shapes. This is one reason to hold that shape
is a type, an ontological category. Another reason is that the notion of shape appears to be
domain-neutral: it is general enough to be applied to astronomy, biology, cognitive science,
geography, geology, physics and psychologyXI. In considering shapes as independently
existing entities we predicate properties of them.
The mind-external world does not, however, present us with these geometric
shapes—these ideal and often perfectly symmetrical entities—as independently existing. This
does not imply that there is no correlation, in some principled sense, between geometric
models (or descriptions) and physical phenomena that are explained by those models.
Although there are no perfect cylinders or other simple shapes, more complex geometric
shapes are found in the world. Fractal geometry has numerous natural manifestations,
including trees, leaves, lightning, and sea shells. When we shift from the abstract space of the
geometer to the physical space of our universe—to determine whether the former explains,
corresponds to, is concretized or manifested in reality—we do not find independently existing
circles or squares. There are, in reality, no isolated points, lines or surfaces.[5] If these entities
are anything over and above mental constructs, they are dependent on other entities for the
existence.
Nature presents us with shapes similar to those in the geometer‘s toolbox, but what
we are presented with is objects with a shape. We recognize shapes in objects, so to speak, and
this is how we pretheoretically (and linguistically) understand shape as a property. A
honeycomb is hexagonal, but not a precisely symmetric hexagon; the orbits of planetary
bodies are represented, explained, and predicted using the conic sections, but bodies and their
orbits undergo perturbations due to gravitational gradients (among other influences); and
planetary bodies exhibit spheroidal and ellipsoidal shapes but are not perfect spheres or
ellipsoids. The list goes on. The more that shapes are specialized, in an ontological hierarchy
for example, the closer these phenomena can be formally represented and described. The
shapes of natural bodies are irregular. These physical or organic shapes do not reflect the
exact specifications of mathematical descriptions of geometric shapes. They approximate
geometric shapes. More precisely, (S4) (the shape of) objects in the world approximate
geometric shapes. Furthermore, category terms such as ‗hexagonal‘ and ‗spherical‘ have a
certain degree of vagueness that reflects the imprecision with which individuals exhibit
geometric shapes. This vagueness, this generality, also permits a variety of physical shapes to
be considered instances, if not types, of the respective term.
We may treat geometric shapes as if they were independently existing entities, but
common sense indicates that they are abstractions with no exact mind-external physical
manifestation, and it would be a mistake to betray that intuition. That which we consider to be
shape is intimately dependent on that which has the shape. In the mind-external world, shapes,
it seems, are properties of things. They [things] must have a shape, i.e. be delineated by a
shape.[6, brackets added] We say that a physical object exhibits a shape. Thus, (S5) shapes
must always be shapes of something in the mind-external world. Outside idealized geometric
space, it does not make sense to posit the existence of an independently existing shape, a
shape with no bearer. The shape cannot exist, but as an idea, without an entity that bears,
exhibits, or has that shape.
XI
With respect to, say visual mental representations.
This does not detract from the theoretical and practical utility of manipulating
geometric entities within conceptual space because it is often in that space that the seeds of
insight (into how the world works) are planted, and it is to the external world that we apply
those insights in order to scientifically progress and bear the fruit of those seeds.
Generally speaking, (S6) shapes are dependent entities. [2] classifies this kind of
entity as a dependent continuant. These dependent entities are said to inhere in (the inherence
relation) objects. What is it to have a shape? Having a shape (a dependent continuant) is to
bear a propertyXII. The inherence relation seems to be what is meant by having a shape. With
regard to the distinction between types and individuals, we have shape property types, and
shape property instances. We can be more precise in the relationship between the shape
property instance and its bearer by explicating the dependence relation between them.
A shape-instance (a physical or organic shape), a, is one-sidedly dependent on its
bearer, b. This means that a is one-sidedly dependent upon b if and only if a is dependent on b
but b is independent of a.[7] Alternatively, the relation between a shape and the object in
which it inheres is that of specific dependence for instances. Specific dependence indicates
that when the (specific) bearer ceases to exist, the (specifically) borne entity (the dependent
entity, the shape instance) ceases to exist as well. As [8] states:
On the instance level… a depends_on b =def. a is necessarily such that if b ceases to
exist than a ceases to exist.
On the type level… A specifically_depends_on B =def. for every instance a of A,
there is some instance b of B such that a depends_on b.
For types, this amounts to saying that all shape instances have some entity (a bearer) on which
they existentially depend. That bearer is most often a material entity such as an object.
Specific dependence is a possible relation between shapes and objects because shape instances
are unique to each bearer. The shape of this particular sculpture is not borne by any other
sculpture, just as the shape of any organism is unique to that organism while exhibiting
attribute-agreement and a similarity of kind. They are similar, but not exact replicas of one
another, and their difference is due to variability in the aforementioned factorsXIII.
Shape instances are not isolable from their bearer. The only sense in which they are
isolable is (a) cognitively, as an abstraction, and (b) in representational artifacts [9], [10].
Physical shapes are more akin to shape instances. Geometric shapes are more akin to shape
types. A star shape, as a geometric shape, has star shape property instances exhibited by, say,
particular star fish. The objects we observe in the world approximate geometric shapes, and
deviate from those shapes by varying degrees. In the next section I explore the more general
notion of form and introduce some definitions of both ‗form‘ and ‗shape‘.
4.0 Form and Shape
In saying that something has the form of a circle, the use of the phrase ―form of a
circle‖ is simply a manner of speaking. Uttering this (and similar) phrase(s) can be understood
as speaking as if the circle is an independently existing entity, having a property: the form.
Our commonplace expressions and knowledge of shapes reflect a primarily qualitative,
XII
XIII
[2] and [20] uses the term ‗quality‘ instead of ‗property‘.
See the end of section 2.0.
perhaps purely visual, understanding of these entities. When we say something has the form of
a circle, square, triangle, or any other geometric shape, we are speaking as though there is an
entity in the world that is a circle and its form—a property of this entity—is its shape. We do
not sincerely and pretheoretically believe this to be the case, however. We know that this kind
of discourse does not make reference to a free-floating shape in space, but involves a turn-ofphrase, an analogy or an abstraction at best. In this context, this is the extent of our
prephilosophical considerations of shape. It is arguable, of course, but our commonsense
intuitions about geometric entities (perhaps all mathematical entities) may be that
mathematical realism does not obtain in the world.
We also utter expressions such as ―That cloud has the form of a bird.‖, where ―form‖,
again, is synonymous with ―shape‖. In this context, we use ―a bird‖ as a general phrase, aware
of the existence of particular birds. Indeed, our notion of shape is intimately connected with
our experience interacting with the world. We most commonly know the form or shape of a
bird by observing individual birds, but this is surely not the only avenue into knowledge of
bird forms. Part of the regularity and attribute-agreement among species of birds (and other
natural or biological entities) is their form or shape. We recognize recurrent shapes that vary
slightly from one another in the world. That there are similar bodily forms (shapes) is
sufficient to posit corresponding types. Moreover, the object in question—the bird—
cognitively takes on a degree of vagueness just as shape terms such as ‗spherical‘ do. That is,
we do not necessarily have a particular type of bird in mind when we utter these expressions,
but do have an approximate and primarily qualitativeXIV idea of the body shape of birds. In
short, ―form of a circle‖ reads ―the shape of a circle‖, which suggests that circles have a
property called ‗shape‘. What is the shape of a circle? The answer, surely, is that it is circular!
We have a shape of a shape, but again, this is a façon de parler.
The term ‗form‘, itself, is particularly ambiguous. It can be used in a broader fashion
than ‗shape‘, and is used in a number of domains of inquiry, such as art, biology (as in
morphology), and philosophy. ‗Form‘ (the Greek eidos) has been defined in multiple ways
[11]:
F1) The shape and structure of something as distinguished from its material
F2) The essential nature of a thing as distinguished from its matter
F3) One of the different modes of existence, action, or manifestation of a particular
thing or substance : KIND
F4) One of the different aspects a word may take as a result of inflection or change
of spelling or pronunciation
F5) A mathematical expression of a particular type
F6) Orderly method of arrangement (as in the presentation of ideas) : manner of
coordinating elements (as of an artistic production or course of reasoning). Or a
particular kind or instance of such arrangement.
F7) A visible and measurable unit defined by a contour: a bounded surface or
volume
These definitions, in some form or other, reflect the ideas of the ancient Greeks, namely
Plato‘s Theory of Forms [12]. The idea common to the definitions is that of the distinction
between generality, repeatability and recurrence on the one hand, and particularity on the
other. In modernity, the former is expressed with the term ‗universal‘ to signify what is
XIV
We do not, for example, intuitively know precise proportions.
universal or commonly shared, general and similar about particular entities in the world.
Generality and similarity among particulars is due to the nomological and structured or
patterned character of the world. That pattern is communicated formally, as (F5) suggests,
with mathematical and physical (as in physicsXV) expressions.
Consider, now, the following definitions and descriptions of ‗shape‘, which provide
some valuable insights, of which I find the first particularly useful.
1) The shape of an object located in some space is a geometrical description of the
part of that space occupied by the object, as determined by its external
boundary – abstracting from location and orientation in space, size, and other
properties such as colour, content, and material composition.XVI
2) External form or contour; that quality of a material object (or geometrical
figure) which depends on constant relations of position and proportionate
distance among all the points composing its outline or its external surface; a
particular variety of this quality.[13]
3) The outward form of an object defined by outline.
The figure or outline of the body of a person.[14]
4) The particular physical form or appearance of something.[15]
The reference to constant relations and proportionality in (2) reflects properties essential to
geometric shapes. For example, that the points on the circumference of a circle are equidistant
from the center reflects constant relations and proportions at work. What is common to most
of these definitions is an external aspect related to shape, a central consideration when seeking
to describe the general features of the category of shape. The shapes of objects (physical
shapes) are identified relative to and based on the external surface, boundary, or form of the
object.
5.0 Boundary, Surface and Shape
The world presents us with numerous physical discontinuities, from the exterior
boundaries of solid objects to interfaces (a type of boundary) such as the interface between the
surface of the ocean and the atmosphere above. Objects, such as an orange, a flower, or a
coffee mug, can be clearly identified as having physical boundaries. The crisp edges of a sheet
of paper delineate the spatial and material extent of the sheet. These boundaries separate, but
do not isolate, objects from one another and from their surrounding environments. The
environment, broadly speaking, and the object are, in fact, interacting (if only at certain
granular levels). They are not isolated from one another. We identify shapes by these physical
discontinuities, and they help us to understand the relationship between material entities and
their shape.
Form, shape, boundary and surface are related. Form can be asserted to be a
superclass of shape, and boundary a superclass of surface. The totality of the surfaces of an
object determines, in part, the shape of the object. In other words, the shape of an object is
determined by its external boundary(s). The surfaces—the external boundaries—of objects, in
XV
It is physics that provides us with the connection to and the comprehension of phenomena in the
world, not pure mathematics.
XVI
The original source is unknown, but can be read at http://en.wikipedia.org/wiki/Shape.
turn, depend on the material constitution of the object, as well as external and internal
processesXVII affording the formation, and preservation, of the boundary. For a very nice
treatment of external and internal processes see [16] and [17].
(S7) All shapes (geometric and physical) have, are composed of, or are bounded by
boundaries.XVIII Shapes are also self-contained. The external boundaries of 1D shapes are the
shapes themselves. The external boundaries of 2D shapes are more similar to outlines or
blueprints, and are a composite of 1D boundaries. The external boundaries of 3D shapes are
surfaces, where surfaces can be described as 2D shapesXIX. Objects end at their surfaces. The
shape of an object—the physical or organic shape—is identified at the limits of the spatial
extent of the object. The shape of an object depends on the spatial extension of the object. If
many objects share similar limits of spatial extension, we say that those objects have the same
shape. In this way, that similarity affords the positing of shape property types. The object,
then, is said to bear a shape property, an instance of the shape type. That instance—the
particular shape—is understood as a property of the object.
Consider a particular vase and the shape of the vase. The shape of the vase is not the
vase. The shape of the vase, considered in itself or in isolation from the vase, is if nothing
more, an abstraction of an aspect of the vase. That aspect—the shape—is perceived as a
gestalt (a whole, a unity or totality) in virtue of the limit of the spatial and material extension
of the vase as well as in virtue of our cognitive apparatus.XX The vase ends at its surface. We
observe discontinuity between the vase and its surroundings. The physical discontinuity
between object and the surrounding medium or space makes the shape of the object apparent
to our awareness. The presence of physical discontinuities affords the existence and
observation of physical shapes.
Thus, we may describe shape by referring to boundaries or physical discontinuities in
reality. A 2D shape, for example, can be understood as the figure formed by, or the region
enclosed by, (a) self-connected bonafide boundaries, or (b) the projection of fiat boundaries
[18] onto an entity or spatial region that is otherwise not delimited. An example of such an
undelineated region would be the spatial regions used in triangulation. The 2D spatial region
traced out from the earth to the moon to the sun is an undelineated or fiat spatial region. With
regard to (b), we do not identify physical or organic shapes in the world, but superimpose or
project geometric shapes onto physical entities, distances or spatial regions as well as entire
phenomena.
In a sense, this is describing fiat shapes, shapes that are demarcated by human
choice. They are shapes delimited in regions that may or may not have existing physical
discontinuities. With the projection of fiat boundaries, we identify fiat shapes. These fiat
shapes are often, if not always, based on an awareness of or familiarity with geometric shapes
and objects exhibiting shapes. You can trace out a circular shape in the air with your finger,
but without the cognitive awareness of the gestalt, the conceptual whole, that is the circle (or
circularity), you are doing nothing other than moving your finger. There is always a cognitive
XVII
Examples include molecular motion, and environmental as well as atmospheric pressures.
Perhaps shapes can be characterized as an abstraction of boundaries.
XIX
I do, however, find it unsettling to posit the literal existence of 2D parts of a 3D entity, specifically
the mind-external existence of 2D surfaces in our 3D space-time.
XX
In describing shape in this way, it should become clearer that there cannot be free-floating shapes in
anything but geometric or conceptual space, and shapes (like all properties) cannot be isolated except in
the mind. This isolation constitutes some acts of abstraction.
XVIII
aspect to shapes. Where no physical discontinuities are present (or subsequently created), only
fiat shapes can come to be.
What is the case, at minimum, is that we mentally project ideal geometric shapes
onto entities in the world, and extract geometric shapes from entities in the world. The
projection of a 2D shape may be an abstraction from a spatial region or an abstraction that is
superimposed onto a spatial region, rather than the spatial region itself. Depending on what
space(-time) is, if anything in itself, the shape is not necessarily the space itself, but is related
to it. When we consider 3D geometric or organic shapes there is a sense in which the shape is
related, if not identical, to the spatial region that the shape bounds. Manipulating geometric
shapes in geometric space presupposes the general notion of space. In observing or thinking
about the shape of individual objects we cognitively extract their shape (an act of abstraction)
as a bounded volume of the space occupied by the object. This extraction ideally has the exact
dimensions of the spatial extension of the object, but this exactness is better suited for
computational applications. In transitioning to the geometer‘s realm, that extraction can
become generalized so that the exact dimensions are excluded, leaving behind a shape type.
3D shapes (and shape in general) are inextricably related to our notion of space, the space we
live and breathe in.
The shape of an object is dependent on the material constitution of the object (among
other factors). Boundaries, in general, exist always in consort with, and are determined in their
nature by the things they bound.[5] The shape of an object is determined by the outer or
exterior boundary of the object. If no physical discontinuity is present, then no (bona fide)
physical shape is present. (S8) The shape of an object is dependent on the object and
determined by the exterior boundary or surface of the object.
A shape is recognizable in the world because we observe similarity in (i) the
boundaries of objects in the world or (ii) in artifacts representing shapes. When the geometer
cognitively manipulates shapes she also (mentally) observes the totality of the external
boundary of the shape. We may also describe shapes in the following manner. To say that
there are shape types or universals in reality that are instantiated as physical shapes in objects
is to say certain natural principles guide and underlie phenomena, processes and entities in the
world. The non-random interaction (or dynamicity) of these entities and the manifestations of
natural principles unfolds in a way that consistently and recurrently results in objects
exhibiting a certain limit to the spatial extension of the object. The totality of that limit is
cognitively perceived as a whole and is considered to be the shape of the object. The
geometric shape of an object, then, is in some sense the abstraction of the limit of the physical
extent of the object. Describing shape in this way—with reference to acts of abstraction—does
not imply that there are no physical boundaries constituting the whole external surface of an
object.
In other words, the shape of material objects is shaped by a number of elements:
natural principles underlying physical processes, the entities that participate in processes, and
the internal processes of objects and other entities. The limit marks a boundary, a physical
discontinuity between the spatial extension of the object and its environment (or other
objects). It is conceived and perceived as a unity, a gestalt. As such it can be recognized time
and again when similar unities (similar shapes) are presented. When an animal recognizes the
silhouette of a predator (or prey) and instinctively flees for safety (or commences the pursuit),
the silhouette is cognitively apprehended as a gestalt. (S9) Shapes are unities, wholes, and are
understood and conceptualized as such.
Finally and in brief, the cognitive element that has been mentioned permits the
introduction of art and representation as they relate to shape. How are shapes related to their
representations, such as artworks? A drawing, a painting, or a computer-generated graphic of a
circle (or any shape-geometric or organic) are each representational artifacts intended to
represent. Assuming the intent of the artist is to represent a circle, the content of the
representation is, surely, a circle but the question remains: what is the ontological status of the
circle, and of shape in general? Is the representation that of an ideal cognitive construct that
nonetheless has some relation to mind-external realityXXI, that of the idea of the circle or of
circularity, that of a shape universal, or a representation of individual circular shaped objects
in the world? Do art works represent a particular circle or the general entity circularity, that is
the Circle type? This is complicated territory, involving mental representation, intentionality,
artifactual representation, perception, and cognition in general. Although beyond the
limitations of this communication, these are relevant considerations for the ontology of shape.
Indeed, the philosophical and applied ontology development of top- mid- or lower-level
ontologies, necessitates, in my view, further exploration into the human element, specifically
intentionality, consciousness and cognition, creativity, artificiality, and the mental domain
overall. Ontologies themselves, after all, are artifacts.
6.0 Granular Shape
Shape has a relationship to granularity. At the mesoscopic granular level of tables,
chairs, trees and people, we can identify imperfect cases of geometric shapes. When we zoom
into the surface of a table we see the cracks, bumps, and imperfections not perceptible with the
naked eye.
The shape of objects will physically change with the change in the material
constitution and boundaries of the object. The shape of objects will also figuratively change
with the granular level at which we analyze the object. At the mesoscopic level of everyday
objects, we identify approximate shapes of things. When we perform a granular descent—
analyzing an entity at a finer level of detail (a finer granular level)— we see that the shape of
these objects deviates from their previously identified geometric shape. The edges of the
aforementioned sheet of paper are crisp (flat, smooth, etc.) relative to the mesoscopic granular
level, but coarse at a finer level. We observe microscopic crevices, indentations, valleys and
other irregularities in the surface, in the material of the sheet. Thus, finer granular levels will
exhibit the irregularities present in the surface.
The exact shape of objects may, in fact, be indeterminate if only due to the margin of
error and finite accuracy of our instruments, including our eyes. These facts seem to mirror
approaching a mathematical limit: we can descend to lower granular levels and get a more
accurate description of the shape without a full characterization of it. Yet when we perform
granular descents there is a sense in which we no longer are describing the shape of the object
in question. The precision with which we describe the shape of an entity will likely depend on
the need or practical application.
7.0 Towards a Shape Ontology
XXI
The relation to mind-external reality may very simply (and poetically) be ourselves—human
beings—and the manner in which we manifest or act upon mind-internal constructions (such as ideas).
Using the traditional endurant or continuant top-level category, and assuming shapes
are, indeed, properties, qualities, or dependent entities, we can formulate an simple ontology
of shapes similar to the hierarchy displayed in Figure 1 (both figures are rendered using [19]).
Given the foregoing considerations, as well as our commonsense intuitions about shape, this
ontological categorization should be fairly comprehensible. Types, classes or universals are
represented by ovals, individuals or instances by rectangles, and relations by labeled arrows.
Note that some types are not necessarily the direct or immediate subtypes of their related
ancestor; I have omitted some potential intermediate types in order to minimize congestion
and foster brevity. The is a (subsumption) relation holds between types: A is a B is
tantamount to A is a subtype of B and is also understood as Every instance of A is an instance
of B. The instance of (instantiation) relation holds between types and their instances. In
addition, the shape types may alternatively be labeled ‗Circular‘, ‗Spherical‘, ‗Triangular‘, and
so on, rather than ‗Circle‘, ‗Sphere‘ and ‗Triangle‘ without any lose of meaning.
Fig. 1: A simple shape ontology with sibling shape types differentiated by dimensionality. Ovals
represent types or classes, squares individuals, and labeled arrows represent relations between types.
Two relations, aside from inherence, between objects and geometric shapes are used
in these diagrams, but both may not be necessary. The has shape relation (hasShape in what
follows) has as relata object types (the domain) and geometric shape types (the range). The
asymmetric, intransitive and irreflexive approximation relation holds between individual
objects and geometric shapes; the relata are object instances (the domain) and geometric shape
property types (the range). The Earth approximates an oblate spheroid, for example.
Alternatively, an object exhibits a shape, or an object exemplifies [21] a shape. I use
‗approximation‘ to emphasize the inexactness of particulars as compared with ideal abstract
forms or shapes. The greater the nesting—the more specialized subtypes of shapes—the more
accurate the representation of the shape of a particular object. The more specialized the shape
types, the closer we come to truthfully saying (for individuals): object O hasShape shape S.
The hasShape relation in figure 1 can be understood as indicating a principled relation
between the type (Planet as a natural kind) and its property type. When formalizedXXII, this
can be understood as ‗All planets have a spheroidal shape‘ or ‗All planets are spheroidal‘. We
can develop expressions such as ‗Earth approximates Oblate Spheroid‘, and ‗Planet hasShape
Spheroid‘, where ‗Oblate Spheroid‘, ‗Planet‘, and ‗Spheroid‘ are types. Let us consider two
specific concerns.
Earlier I mentioned that shape cannot be categorized as an independent continuant in
the sense of [2] because identity is not preserved with the loss of a part (a side for shapes XXIII)
of the entity (the shape). So long as the parent class—continuant—continues not to be defined
in a similar manner, shapes can remain as dependent continuants, and thus fit within the
hierarchy. If all continuants preserve their identity in the face of the loss or acquisition of
parts, then it is not clear that shapes do so, but [2], displayed in [22], cites shape as a quality (a
dependent continuant). This will call for inquiry into parthood as it relates to shape. Since
geometric shapes may be purely mathematical entities a mid-level ontology of these kinds of
entities would also be worth exploring. The second concern is that figure 1 neither represents,
nor mentions, shape instances.
If we include shape property instances then we have the following. Appended
numbers indicate unique instances.
Fig. 2: Shape instances with objects and geometric shapes.
The oblate spheroid property instance is exhibited or borne by the Earth. The Earth‘s
particular shape instance inheres in the Earth. Since Oblate Spheroid is subtype of Spheroid
and all planets have a spheroidal shape, the Earth having a spheroidal shape logically follows.
More specifically, the expression ‗Earth hasShape oblate spheroid 1‘ satisfies a constraint
expressing the physical principle that all planets (must) have a spheroidal shape. Similarly,
additional instance expressions can be added, such as ‗Mars hasShape oblate Spheroid 2‘. The
geometer or, more appropriately, the astrophysicist‘s geometric descriptions of the particular
shape of each planet can then be related to the expression of the shape instance.
Here, geometric shapes are considered properties, but it may very well be that shapes
are more appropriately categorized as purely mathematical entities considered as other than
properties. This would justify the introduction of a mathematical entity ontology as a potential
XXII
XXIII
Using sorted or typed first-order predicate logic, for example.
For curved shapes, it is less clear what their parts would be.
lower-level ontology. If geometric shapes are placed within a mathematical entity ontology,
then at least two questions still need to be answered: (Q1) What is the ontological status of
mathematical entities; are they cognitive or mental constructs, representational artifacts,
independent continuants, dependent continuants, immaterial endurants, etc.? How does this
lower-level ontology fit into top-level ontologies? (Q2) How do these entities relate to mindexternal entities such as material objects and the shape of objects? If mathematical entities are
categorized as cognitive or mental entities (perhaps cognitive artifacts), then, in the light of
(Q2) we would need to answer how these cognitive entities relate to natural phenomena and
concrete particulars. In this framework, to say the Earth has the shape of an oblate spheroid is
not to say that the Earth has the shape of a particular thought, concept, or idea, but that the
Earth exhibits certain physical relations that are represented by content of the thought, if not
simply the thought itself. This is a plausible avenue.
Finally, if we wish to include the shape of entities from specific domain such as
astronomy, biology or geography we can include lower-level category terms such as
‗Biological Shape‘XXIV, or ‗Astronomical Shape‘. These, however, seem more questionable in
that they may be explained in terms of other shapes.
8.0 Closing Remarks
I have presented a broad exploration of shape, including its dimensionality and
relation to material objects, form, boundary, surface and granularity in an attempt to
understand what shapes are. It has been suggested that shape is a property—a dependent
entity. As a property, objects can be said to bear, have, or exhibit a shape. More precisely, I
suggested that the shape of an object is determined and identified by the exterior boundary of
the object. The exterior boundary of an object is the limit of the spatial or material extension
of the object. With shape considered as the product of an abstraction, the inquiry becomes
more complex and necessarily involves a cognitive aspect. I introduced the question of the
status of shapes as mathematical entities, and as cognitive devices, discussing artifactuality,
and abstraction, specifically projection of shapes and extraction of them.
The ontology of the canonical world of geometry is one of perfect entities: geometric
shapes that are often symmetrical and lacking of distortions or irregularities. Their properties
are often, if not always, described mathematically. Perfect shapes are only found in the
abstract, whereas the mind-external world presents us with irregular shapes, some of which
closely approximate the objects in the geometer‘s toolbox. Our environments never present
shapes in isolation from other entities. We do not experience shapes in isolation from what has
the shape. The only sense in which shapes can be separated from material objects is in the
sense of an abstraction, where the shape is abstracted away from the object. We perform an act
of abstraction such that the shape is conceived as though it were isolable.
The distinction between types (classes or universals) and instances (individuals or
particulars) paralleled, to a certain degree, the distinction between geometric shapes—ideal,
abstract and perfect shapes that are often mathematically formalized—and physical or organic
shapes, the shape of mind-external entities. Although the shape of objects can be formally
expressedXXV, geometric space presents us with a set of entities (shape types) that are general
XXIV
An example would be the shape of a bird.
For any physical object it is likely that we can formulate a mathematical description and produce an
idealized shape for it. Consider the Earth and the construct of the geoid. An additional characterization
XXV
enough to have objects in mind-external reality approximate if not bear instances of them.
These ideal constructs, so conceived, are similar to types, and shapes of particular objects in
the world are more like instances. Shape, as an ontological category, can be conveyed in terms
of shape property types with specializations thereof, and shape property instances inhering in
individual objects. As more and more specializations of a shape type are discovered or
formulated, we come closer to an expression of the actual shape of things.
An overarching consideration and implicit assumption was that of describing types or
universals using the nomological aspect of the world. For example, if instances (objects) of a
particular kind have a similar shape, they do so in virtue of physical principles underlying the
numerous factors (influences, processes, entities and phenomena in general) that go into
forming the limit of the spatial extension of the object. Some of those elements partially
characterize the object kind.
Finally, I presented a preliminary ontological hierarchy using the approximation
relation and the hasShape relation (section 7). I introduced two potentially overlapping
possibilities (among many, no doubt): one, that geometric shapes are cognitive artifacts, and
two, that geometric shapes belong in a mathematical entity ontology. With these possibilities
in mind I stressed the need to determine the ontological status of shapes as such, as well as the
relation of mathematical entities to (a) objects exhibiting a shape and (b) more generally, to
the phenomena in the world that mathematical and physical expressions purport to explain.
Continued research into the ontology of shapes will, thus, necessitate entering the
domain of the philosophy of mathematics. What is relationship between geometric entities
such as shapes (and mathematical entities in general), the human being, cognition and the
universe? Our use of geometry and physics appears to indicate that our idealizations in
conceptual space are able to touch on (express, explain and discover) aspects of, and
phenomena in, the world. Perhaps ideal shapes such as the Platonic Solids do, indeed, have
efficacious correlations to patterns and processes in the world. There is, indeed, a harmony of,
and in, the world.
Geometry itself will be invaluable to the inquiry, as will development and
formalization of shape axioms and formal relations between the relevant categories. Overall,
clarifying the distinction between abstract ideal shapes and physical shapes is needed.
Additional topics of interest include a mereological (or mereotopological) description of
shapes, the ontology of complex natural shapes such as fractals, the relation between shapes
and negative entities (such as holes), and the shape of temporally-extended entities.
The development of an ontology of shape must take a multi-disciplinary approach.
With the advent of modern science, geometry (and physics) and physical space-time are
interrelated. The former is primarily a mind-dependent enterprise with practical applications
beyond the mind. The latter is understood as mind-independent reality that has a correlation to
the expressions in the former.XXVI Given this overlap, domain-experts from various
disciplines, including artists, cognitive scientists, geometers, mathematicians, and physicists
will positively contribute to inquiry into and understanding of shape.
of shape is this: that two objects have the same shape, or that their particular shapes are instances of the
same shape type means that the external boundary marking the limit of the spatial extension of each of
the two objects can be similarly geometrically described. These objects will not have identical
dimensions, and they will not necessarily match the specifications of the geometric shape that is
considered to be the type of which their shape properties are instances. They approximate the shape type.
XXVI
This too, is an assumption: that modern space-time theories are accurate representations of the
world.
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